Aspects of the General Theory of Relativity

Transcription

1 Aspects of the General Theory of Relativity Chapter IV 1. How does gravity act 2. Cosmological redshift 3.Gravitational redshift 4. Black holes

2 General Relativity: Gravity and the Curvature of Space A light beam will be bent either by a gravitational field or by acceleration (outside observer): Definition of a straight line; The line that a light ray follows

3 An outlook on General Relativity GR deals with: Gravitation Acceleration Principle of equivalence: it is impossible to distinguish a uniform gravitational field and a uniform acceleration. Another way to put it: mass in Newton s first law is the same as the mass in the universal law of gravitation. F = ma F = G 2 Mm R

4 General Relativity: Gravity and the Curvature of Space This can make stars appear to move when we view them past a massive object: Note: The bending of light can in principle be explained by Newtons law (Soldner in 1801) The difference is quantitative; a factor of 2, measured by Eddington in 1919.

5 Gravity and the Curvature of Space Gravitational lensing This bending of light as it passes a massive object (star or galaxy) has been observed by telescopes: Fermat s principle in optics: light traveling between points chooses the shortest track

6 Gravity and the Curvature of Space Einstein s general theory of relativity says that space itself is curved hard to visualize in three dimensions! This is a two-dimensional space with positive curvature: Not known what the overall curvature of the universe is (but close to zero) NB; most curvature is local!

7 Gravity and the Curvature of Space Space is curved around massive objects: Fundamental notion of GR: gravity is not a force but deformation of space calculation is difficult, because of non-euclidean geometry

8 Cosmological Red Shift λ 0 λ z λ 1+ z 0 λ Scale factor: Cosmological redshift Edwin Hubble Galaxies moving away from each other Expansion of the universe Interpretations: Change of the underlying metric in expanding universe Interpretation as a Doppler shift Redshift does not have dispersion Note: z z(λ) T Redshift & time 1 1 = T0 3/ 2 ( 1+ z abs )

9 Expansion of the Universe Hubble measurements H = 71 km/s/mpc = 22 km/s/mly Parsec parallax Earth-Sun is 1 1 Light year = 3 x 10 8 x 3600 x 24 x 365 = 9.5 x m H = 22 x 10 3 / 10 6 x 9.5 x s -1 = 2.3 x s -1 1/H = 1 / 2.3 x s -1 = 4.3 x s = 4.3 x / 3600 x 24 x 365 = 13.6 x 10 9 yr

10 Expansion of the Universe Hubble measurements H = 71 km/s/mpc = 22 km/s/mly How much time did it take for galaxies to be separated at distance d Assuming they depart with Hubble speed v = Hd t = d v = d Hd = 1 H This corresponds to ~13.6 billion years Note: Now (21) Gyrs (Planck Collab.) Accelerated expansion of the Universe

11 Copernicus and Cosmological Principle 1) The Earth is not the centre of the world (solar system) 2) The Sun is not the centre of our Galaxy 3) Universe is - isotropic (looks the same in all directions) - homogeneous(all locations are equivalent)

12 Big Bang and Cosmological Principle Expansion is the same from all points in the Universe Our place is not special Big Bang happened everywhere

13 Gravitational Red Shift In General Relativity it is time that depends on the gravitational dependence. This is at the heart of an explanation of gravitational redshift it is a gravitational Time dilation. But it can be understood as an energy loss in a gravity field. V ( R) = G Mm photon At the surface of a heavy object V ( ) = 0 hν ( g = G ) 2 R and M R M R Photons loose energy when traveling uphill E = h ν = mphotong R Photons shift their energy to the red

14 Red Shifts Doppler effect Gravitational redshift ν = m photon h g R Cosmological redshift λ 1+ 0 λ z

15 Simple (Newtonian) view on a Black Hole R s Escape from a distance R s with an escape velocity v = R s = V e 2GM R 2GM c 2 Requirement: Kinetic energy must beat the gravitational potential mv = GmM R Take c for the escape velocity (of course not correct but some approx) Schwarzschild radius (also valid in GR) Laplace (1795): possibly the greatest luminous bodies are invisible

16 Intermezzo The spectrum of a quasar

17 Intermezzo Search for varying constants in the early Universe Compare the absorption spectrum of H 2 in different epochs Lab today QSO 12 Gyr ago Each line is redshifted λ λ z i 0 i 1+ z i nm ~ nm Spectral lines of a molecule depend on p the fundamental constant of nature : µ = = (75) M m e

18 Intermezzo Search for varying constants in the early Universe Laboratory spectra For z=0 Astronomical spectra For high z Make a comparison

19 Intermezzo Quasar Q ; the most distant At z abs = 4.22 ; 1.5 Gyrs after the Big Bang spectrum z λi 0 λ i µ µ ( 1+ z ) 1+ K abs i Important: Knowledge from Molecular Physics K i values different for all spectral lines Molecules are sensitive for the fundamental constants µ µ Result <

20 A Stringent Limit on a Drifting Proton-to-Electron Mass Ratio from Alcohol in the Early Universe Bagdonaite, Jansen, Henkel, Bethlem, Menten, Ubachs, Science 339 (2013) 46 Intermez Effelsberg Radio Telescope PKS molecular factory K=-33 at z= (7.5 Gyrs look-back) K=-1 K=-7 µ <10 7 µ

21 Intermezzo Anthropic Ideas on the Universe α = (96) µ = m m p e = (85) No theories for the values of these parameters underlying structure of matter in the universe See the paper: Carr & Rees, Nature 278 (1979) 605

22 Intermezzo The Hoyle Resonance Fred Hoyle